Did you try the Khun & Tucker algorithm?

Kuhn, H. W.; Tucker, A. W. Nonlinear Programming. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 481--492, University of California Press, Berkeley, Calif., 1951. http://projecteuclid.org/euclid.bsmsp/1200500249.

Thank you sir.

Hi Kerim,

In addition to Aliyeh Kazemi's answer CPLEX is only able to solve the NLP's in "quadratic form!".

If you try to solve an NLP optimally, you have to check the optimally conditions first. But as i know most of the time its hard to check the optimality conditions for an NLP. Even you cannot solve some nonlinear equation systems analytically. That's why there are numerous methods proposed in the literature, such as "Newton Method", "Secant Method", "Uniform Search", "Dichotomous Method", "Golden Section Method", "Fibonacci Method", "Root Mean Square Method" etc...

In addition to this there is a solver called Baron. Its an commercial software. I don't know whether it support academic licensing. And as i know it gives good quality solutions to almost all type of NLPs.

Best regards.

Dear Kerim,

There are several good books about the subject.

My favourites are:

@book{mybib:Fletcher,

AUTHOR = {R. Fletcher},

TITLE = {Practical Methods of Optimization},

YEAR = "1987",

edition = "2nd",

PUBLISHER = {John Wiley, Chichester}

}

@book{mybib:NocedalWright,

author = "J. Nocedal and S.~J. Wright",

title = "Numerical Optimization",

year = "1999",

publisher = "Springer-Verlag Telos",

address = "Berlin"

}

There are various software packages for NLP, some of them are certainly hooked to modelling languages such as AMPL or GAMS.

For example: GALAHAD, FilterSQP, SNOPT, IPOPT, KNITRO.

You'll need to find their websites.

Depending on the type of nonlinearities in your problem some of these codes may perform amazingly well or struggle a bit.

This is because NLP problems (even small ones) might be difficult.

With best regards,

Jacek

Soft computing methods like Genetic algorithm, Tabu search etc are also provide best result for solving Nonlinear programming problem.

Thank you for your valuable answers. I advise a web site to researchers who are interested in GAMS...It includes many NLP, LP, IP or MIP samples...

https://www.gams.com/modlib/libhtml/subindx.htm

Best regards,

You should start by analyzing the problem - no need to resort to heuristics before you have ruled out better ways. Check if the function f is convex, the functions g_i involved in the inequalities are concave, and the functions g_i involved in the equalities are affine. If that is the case then you have a convex problem, for which there are plenty of algorithms that will guarantee that you find a global optimum. If not, then such algorithms can still provide fairly good approximate solutions if you test starting from different points - some choices might be natural because you might know something about the model, such as whether some variabels ought to have large or low values, and so forth. If your functions are also differentiable than you have access to good approximations from first- or second-order approximations, whence Newton-like methods can be applied. In any case, do try "conventional" nonlinear programming methods before you resort to some heuristic means.

meta-heuristic methods such as Genetic Algorithm (GA), Particle Swarm Optimization etc. can be utilized for nonlinear programming. However, using Non-Linear Programming is the best way. They can be implemented in Matlab software.

I fully agree with Michael's answer. What you might get depends on the properties of your NLP. For non-convex problems things get much more difficult if one really wants a solution close to a global optimum. Then something like domain decomposition might help (e.g. BARON solver in GAMS). But there is no guarantee of a fast solution especially in higher dimensional problems and with wide variable bounds.

GAMS itself is no solver, but a modelling language/software - a compiler for optimization models - with interfaces to different solvers.

If you have GAMS available then you got a good choice of the mentioned NLP software packages to try.

But  CPLEX advocated in one of the answers above is not among the solvers for general NLPs - it is good for MILPs  and can also solve MIQPs: mixed-integer QUADRATIC problems.

Depending on the GAMS license you have the licensed solvers can be used to solve larger problems. At least you can try every contained NLP/DNLP (not continuposly differentiable) softtware in demo mode (i.e. on small problems).

What solvers are avalable and for which type of problem you see in the documentation

http://www.gams.com/help/index.jsp

and which ones are licensed in your installation you find out by calling gamsinst as root (in a Unix/Linux environment). To try different solvers on the same model call GAMS with the command-line option naming the solver to be used, e.g.

gams  myproblem.gms  NLP=IPOPT  O=ipopt.lst Lo=4 Lf=ipopt.log

gams myproblem.gms NLP=CONOPT O=conopt3.lst Lo=4 Lf=conopt3.log

gams myproblem.gms NLP=CONOPT4 O=conopt4.lst Lo=4 Lf=conopt4.log

gams myproblem.gms NLP=SNOPT O=snopt.lst Lo=4 Lf=snopt.log

By setting Lo=4 you get a log output to the screen AND a file, setting the O and Lf command line parameters to different names prevents overwriting of the log and lst files so that you can easily compare results afterwards.

My experience showed: it might help for nonconvex problems where even finding a feasible point is hard for the algorithms to first call one type of solver e.g. CONOPT and write the variable values to an external file by the GAMS put facility. If it succeeded, o.k., else use the final point as starting values with a different type of algorithm, e.g. IPOPT.

The book "Convex Optimization" by Boyd and Vandenberghe is good too.

Thank you for contributions. While searching net, i found a chapter from MIT web site about NLP. I want to share with you.

Applied Mathematical Programming by Bradley, Hax, and Magnanti (Addison-Wesley, 1977)

For a better introduction than the chapter you found I would recommend

Bazaraa/Sherali/Shetty: Nonlinear programming, Wiley (different editions over the years)

The nonlinear programming tasks require the presence of a non-linear objective function and nonlinear constraints. Application of regular search methods to such problems associated with certain difficulties. I would advise to use the stochastic programming techniques.

There are methods that are symbiosis of stochastic and deterministic methods.

We recently published a paper on solving nonlinear MIP problems under the monotonic condition.  To solve such problems, we used the combination of surrogate Lagrangian relaxation and branch-and-cu to relax selected coupling constraints. Monotinicity of the relaxed problem is then exploited and nonlinear terms will be dynamically linearized. The linearity of the resulting problem will be exploited using branch-and-cut. For your perusal, the paper on surrogate Lagrangian relaxation and the combination of surrogate Lagrangian relaxation and branch-and-cut are attached.